The Theory And Practice Of The H P Version Of Finite Element Method

The Theory and Practice of the H-p Version of Finite Element Method

There are three versions of finite element method. The classical h-version achieves the accuracy by refining the mesh while using low degrees P of elements, p=1,2 usually. The p-version keep the mesh fixed and the accuracy is achieved by increasing the degree p. The h-p version properly combines both approaches. The h-p version is the new development of finite element method. It was first addressed by Babuska and Dorr?4 . The further analysis and computation for two dimensional problems were made by Guo, Babuska where the exponential rate of convergence was proved. The one dimensional analysis was given by Guo Babuska. The improvement of the results for curvilinear boundary and curvilinear elements was made by Babuska, Guo. The problem with non-homogeneous Dirichlet data was studied by Babuska, Guo. The h-p version with elements for the problem of 2m order was discussed by Guo. The feedback and adaptive approach was developed by Guo, Babuska and Babuska, Rank. This paper is addressing some theoretical advances and presents numerical illustrations.

Scientific and Technical Aerospace Reports

hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

Over the last few decades discontinuous Galerkin finite element methods (DGFEMs) have been witnessed tremendous interest as a computational framework for the numerical solution of partial differential equations. Their success is due to their extreme versatility in the design of the underlying meshes and local basis functions, while retaining key features of both (classical) finite element and finite volume methods. Somewhat surprisingly, DGFEMs on general tessellations consisting of polygonal (in 2D) or polyhedral (in 3D) element shapes have received little attention within the literature, despite the potential computational advantages. This volume introduces the basic principles of hp-version (i.e., locally varying mesh-size and polynomial order) DGFEMs over meshes consisting of polygonal or polyhedral element shapes, presents their error analysis, and includes an extensive collection of numerical experiments. The extreme flexibility provided by the locally variable elemen t-shapes, element-sizes, and element-orders is shown to deliver substantial computational gains in several practical scenarios.